Almost sure convergence of weighted sums of mixing sequences. (Chinese. English summary) Zbl 0869.60029

Yang, Shanchao

J. Syst. Sci. Math. Sci. 15, No. 3, 254-265 (1995).

For a weighted sum of a sequence of i.i.d. random variables having finite second moment, under the conditions that the weight matrix is lower diagonal and the sum of squared weights equals 1, Y. S. Chow [Ann. Math. Stat. 37, 1482-1493 (1966; Zbl 0152.16905)] first proved its strong convergence. This result was later generalized by R. Thrum [Probab. Theory Relat. Fields 75, 425-430 (1987; Zbl 0599.60031)] to i.i.d. random variables with finite \(p\)th moment \((p\) greater than 2). The author considers the strong convergence of the weighted sum of a sequence of mixed random variables. Theorems similar to that proved by Chow and Thrum are established under weaker conditions. Especially, the sum of the squared weights is no longer required to equal 1. Applications of these results to nonparametric regression estimates are mentioned, but no detail or reference are given.

Reviewer: D.Tu (Kingston)

MSC:

60F15	Strong limit theorems
62J05	Linear regression; mixed models

Keywords:

weighted sum; strong convergence of the weighted sum; nonparametric regression estimates

Citations:

Zbl 0152.16905; Zbl 0599.60031